White noise analysis
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In probability theory, a branch of mathematics, white noise analysis, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process.[1] It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes.[2]
The term white noise was first used for signals with a flat spectrum.
White noise measure
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Perspective
The white noise probability measure on the space of tempered distributions has the characteristic function[3]
Brownian motion in white noise analysis
A version of Wiener's Brownian motion is obtained by the dual pairing
where is the indicator function of the interval . Informally
and in a generalized sense
Hilbert space
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Perspective
Fundamental to white noise analysis is the Hilbert space
generalizing the Hilbert spaces to infinite dimension.
Wick polynomials
An orthonormal basis in this Hilbert space, generalizing that of Hermite polynomials, is given by the so-called "Wick", or "normal ordered" polynomials with and
with normalization
entailing the Itô-Segal-Wiener isomorphism of the white noise Hilbert space with Fock space:
The "chaos expansion"
in terms of Wick polynomials correspond to the expansion in terms of multiple Wiener integrals. Brownian martingales are characterized by kernel functions depending on only a "cut-off":
Gelfand triples
Suitable restrictions of the kernel function to be smooth and rapidly decreasing in and give rise to spaces of white noise test functions , and, by duality, to spaces of generalized functions of white noise, with
generalizing the scalar product in . Examples are the Hida triple, with
or the more general Kondratiev triples.[4]
T- and S-transform
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Perspective
Using the white noise test functions
one introduces the "T-transform" of white noise distributions by setting
Likewise, using
one defines the "S-transform" of white noise distributions by
It is worth noting that for generalized functions , with kernels as in ,[clarification needed] the S-transform is just
Depending on the choice of Gelfand triple, the white noise test functions and distributions are characterized by corresponding growth and analyticity properties of their S- or T-transforms.[3][4]
Characterization theorem
The function is the T-transform of a (unique) Hida distribution iff for all the function is analytic in the whole complex plane and of second order exponential growth, i.e. where is some continuous quadratic form on .[3][5][6]
The same is true for S-transforms, and similar characterization theorems hold for the more general Kondratiev distributions.[4]
Calculus
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Perspective
For test functions , partial, directional derivatives exist:
where may be varied by any generalized function . In particular, for the Dirac distribution one defines the "Hida derivative", denoting
Gaussian integration by parts yields the dual operator on distribution space
An infinite-dimensional gradient
is given by
The Laplacian ("Laplace–Beltrami operator") with
plays an important role in infinite-dimensional analysis and is the image of the Fock space number operator.
Stochastic integrals
A stochastic integral, the Hitsuda–Skorokhod integral, can be defined for suitable families of white noise distributions as a Pettis integral
generalizing the Itô integral beyond adapted integrands.
Applications
In general terms, there are two features of white noise analysis that have been prominent in applications.[7][8][9][10][11]
First, white noise is a generalized stochastic process with independent values at each time.[12] Hence it plays the role of a generalized system of independent coordinates, in the sense that in various contexts it has been fruitful to express more general processes occurring e.g. in engineering or mathematical finance, in terms of white noise.[13][9][10]
Second, the characterization theorem given above allows various heuristic expressions to be identified as generalized functions of white noise. This is particularly effective to attribute a well-defined mathematical meaning to so-called "functional integrals". Feynman integrals in particular have been given rigorous meaning for large classes of quantum dynamical models.
Noncommutative extensions of the theory have grown under the name of quantum white noise, and finally, the rotational invariance of the white noise characteristic function provides a framework for representations of infinite-dimensional rotation groups.
References
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